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A Neumann boundary-value problem on an unbounded interval. (English) Zbl 1173.34318
Summary: We study the Neumann boundary-value problem on the half line:
\[ y''(x)=f(x,y(x),y(0),y(\infty)),\quad x\in(0,+\infty), \]
\[ y'(0)=v_0,\quad y'(\infty)=0, \]
where
\[ y(\infty):=\lim_{x\to+\infty}y(x),\quad y'(\infty):=\lim_{x\to+\infty}y'(x) \]
and \(f:[0,+\infty)\times \mathbb R^3\to\mathbb R\) is continuous. An existence result is obtained by an adapted version of the method of upper and lower solutions, together with a diagonal argument.

MSC:
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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